=Paper=
{{Paper
|id=Vol-3919/short2
|storemode=property
|title=Indoor Position Anti-jam Via Robust IPNCM
|pdfUrl=https://ceur-ws.org/Vol-3919/short2.pdf
|volume=Vol-3919
|authors=Lichao Gao
|dblpUrl=https://dblp.org/rec/conf/ipin/Gao24
}}
==Indoor Position Anti-jam Via Robust IPNCM==
https://ceur-ws.org/Vol-3919/short2.pdf
Indoor Position Anti-jam Via Robust IPNCM
Lichao Gao1,∗
1
Beijing Research Institute of Telemetry, Beijing No. 1 Nanda Hongmen Road, China
Abstract
Indoor position in interference and multipath environment need adaptive beamforming to realize
interference filtering and navigation signal demodulation. However, module mismatches include steer
mismatch and array mismatch occurs frequent and will cause adaptive beamforming performance serious
degradation. Using uncertainty set to constrains mismatch error is robust but the set size is hard to
decide. Aim at this problem, we propose a novel robust IPNCM type algorithm without any uncertainty
set constrains but mismatch fix matrixes. The key idea of the new algorithm is constructing a maximum
SINR optimal problem then using an iterative direction set to solve it. By using the verification data in
the signal of interested, the SINR can be estimated. By analyze the subspace character, a finite iterative
direction set can be found, and the NP hard maximum SINR optimal problem can be solved. Unlike most
of previous algorithms, the proposed algorithm is much more robust to module mismatches as it adaptive
achieve fix matrixes to reduce the whole estimation error include the interference steer vector and the
signals of interested steer vector. Numerical results verified that the new algorithm is much more
robustness to large mismatch error to the others.
Keywords
robust adaptive beamforming, IPNCM, module mismatch, gain and phase error 1
1. Introduction
Indoor position via wireless electromagnetic signal like communication, MIMO all experience
performance decline in interference and multipath environment. Adaptive beamforming which
adaptive forming beam in the signal of interested (SOI) direction and forming null in the
interference direction is a classic research topic in array signal processing. It has been widely
applied in mobile communication, and MIMO [1, 2]. While in engineering application, DOA
mismatch of SOI caused by module mismatch will lead adaptive beamforming performance decline,
especially when the SNR of SOI is large than 0, the SOI will also be filtered out.
To solve the mismatch problem, lots robust adaptive beamforming algorithm have been
proposed, but few are suitable to mismatch caused by failed sensors in large array.Dialog loading is
always robust to any kind of mismatch by suppress the target signal lower than the adding noise
[3, 4, 5]. However, the loading factor is hard to choose in engineering. And the dialog loading
algorithms are always loss interference filtering performance. Linear and/or quadratic constrains
adaptive beamforming will achieve robust performance to module mismatch by adding extra
constrains. This kind of algorithm is target at mismatch caused by steer mismatch or array
mismatch which usually are small mismatch [6, 7, 8, 9, 10, 11, 12]. Weight constrains algorithm as
robust to outliers as dialog loading, and have the same drawbacks include the constrains are hard
to choose and may loss interference filtering performance as in the weight constrains may not exist
a global optimal solution [13, 14]. GSC (Generalized sidelobe cancellation) using blocking matrix to
obtain the pure interference-noise signals. So, GSC will received robust beamforming performance
∗
Corresponding author.
erduo411@sjtu.edu.cn (L. Gao)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
while the SNR of SOI is large than 0 [15, 16]. However only if the blocking matrix designed robust
enough to steer mismatch and array mismatch, the SOI will not pass though blocking matrix and
will not be filtered out which is hard to realize [17, 18]. IPNCM (interference-plus-noise covariance
matrix reconstruction) algorithm robust to the mismatch by rebuilding pure interference-noise
matrix based on interference DOA estimation. The module mismatch includes DOA error, array
sensor gain and phase error will lead both direction steer vector estimation error of SOI and
interference. Both steer vector errors will lead the beamformer performance degrade seriously [29].
By Assuming the true direction steer lies entirely in IPNCM estimated, the true steer vector can be
estimated by calculate the intersection of signal subspaces eigen decomposition from array data
covariance matrix and IPNCM. Then the estimation error can be departed into 2 subparts: one is
perpendicular to signal subspace part, and anther is parallel to signal subspace part. Aim to reduce
the estimation error effects, lots robust IPNCM algorithms have been proposed in recent years [19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. However, most of these algorithms
only address the parallel to signal subspace part.
The problem studied in this paper is to realize a robust IPNCM type algorithm to both steer
vector error parallel to signal subspace parts and perpendicular to signal subspace parts. We show
in this paper that by using verification data in SOI, a maximum SINR optimal problem can be
constructed and solved by interactive methods in section IV. And we prove that, the algorithm
converges to optimal solution with no extra assume. In section V, the proposed algorithm is
compared with LCMV, Worst-Case, IPNCM, VSP-IPNCM proposed in [23], RCB-IPNCM proposed
in [27], and the robust IPNCM algorithm proposed in [29]. The simulation results shown that the
proposed algorithm achieve a much more robust performance than the others.
2. Problem statement
To set up the problem, consider a scenario where K + 1 far-field narrowband source signals
impinge upon an M-sensor linear array from directions-of-arrival (DOAs) {θk , k = 1, … , K + 1}.
Without loss of generality, the last source signal is taken as the SOI, while the remaining K signals
are regarded as the interferences. The array sensor collected data at time t is denoted by the
following M × 1 vector:
�(𝜃𝜃
𝒙𝒙(𝑡𝑡) = 𝒂𝒂
�� 𝑠𝑠 )s(𝑡𝑡)
��� � 𝒊𝒊(𝑡𝑡) + 𝒏𝒏(𝑡𝑡)
�� + 𝑨𝑨
��� (1)
≜𝒙𝒙𝑠𝑠 (𝑡𝑡) ≜𝒙𝒙𝑖𝑖 (𝑡𝑡)
where θs = θK+1 , a� (θs ) represents the M × 1 practical (mismatched) steering vector of the SOI,
s(t) represents the signal waveform of the SOI, A � = [a� (θ1 ), … , a� (θK )] represents the M × K
mismatched array response matrix of the interferences, a� (θk ) represents the M × 1 mismatched
steering vector of the kth interference, i(t) = [i1 (t), … , iK (t)]T is the K × 1 waveform vector of the
interferences, and n(t) = [n1 (t), … , nM (t)]T is the M × 1 additive noise vector. (∗)T is transpose.
Here, the array mismatch is modeled by unknown gain-phase uncertainties of sensors. In this
case, the mismatched steering vectors can be expressed by left multiplying an M × M complex
diagonal matrix G to the corresponding ideal steering vectors, i.e.,
�(𝜃𝜃) = 𝐆𝐆𝒂𝒂(𝜃𝜃)
𝒂𝒂 (2)
where a(θ) represents the M × 1 steering vector of the fully calibrated array with respect to a
source signal at DOA θ. For a linear array, a(θ) is given as
2𝜋𝜋
𝒂𝒂(𝜃𝜃) = [ 𝑒𝑒 −𝑗𝑗 𝜆𝜆 𝑑𝑑1𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ⋯
2𝜋𝜋 2𝜋𝜋 𝑇𝑇
𝑒𝑒 −𝑗𝑗 𝜆𝜆 𝑑𝑑𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ⋯ 𝑒𝑒 −𝑗𝑗 𝜆𝜆 𝑑𝑑𝑀𝑀𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ]
(3)
where dm represents the location of the mth sensor.
The present problem is to find an M × 1 weighting vector w such that the signal to interference
and noise ratio (SINR) is maximized. Toward this end, the following assumptions are made:
1. The source DOAs are pairwise distinct, i.e., θk ≠ θl , ∀k ≠ l.
2. The value K + 1 < M is correctly determined.
3. The source signals are uncorrelated from each other.
The additive noise is complex white Gaussian and is uncorrelated from the source signals.
3. Robust IPNCM algorithm via DA
Indoor position via communication, MIMO all usually using a set of training data known to both
transmitter and receiver to complete data synchronization [38, 39], which defined as closed-form
data aided (DA) scenario [40].
In interference and multipath environment, first using traditional IPNCM algorithm to improve
signal receive performance. The signal after IPNCM process can be written as:
� 𝒊𝒊(𝑡𝑡) + 𝒏𝒏(𝑡𝑡))
𝑦𝑦𝑠𝑠 (𝑡𝑡) + 𝑦𝑦𝑖𝑖 (𝑡𝑡) = s(𝑡𝑡)e−jθ𝑠𝑠 + 𝒘𝒘𝑯𝑯 (𝑨𝑨 (4)
Then the ML algorithm [41] can use the data aided to estimate the time delay τ and phase delay
θs from transmitter to receiver. The corresponding logarithmic likelihood formula is written as:
�
(5)
𝑳𝑳(θ�𝑠𝑠 τ�) = 𝑹𝑹𝑹𝑹 � 𝐰𝐰 𝑯𝑯 𝐱𝐱(t)s𝑣𝑣𝐻𝐻 (t + τ�) ejθ𝑠𝑠 𝑑𝑑𝑑𝑑
𝑻𝑻
In (5), the 𝑹𝑹𝑹𝑹 is take real part from complex data operation, T is sampling interval. Define the
data aided is 1 × N row vector sv , N is the data length and the typical value is 200[64].Then the
logarithmic likelihood formula can be changed as
𝑳𝑳(θ�𝑠𝑠
� (6)
τ�) = 𝑹𝑹𝑹𝑹 � 𝐰𝐰 𝑯𝑯 𝐱𝐱(n)s𝑣𝑣𝐻𝐻 (n + τ�)ejθ𝑠𝑠
After time delay τ and phase delay θs estimated, using data aided sv can filter out the data
�
aided received by antenna svH (n + τ�)ejθs . Because the data aided is uncorrelated with the
interference signals and noise, so the filter out data aided process will not distort the interference
signals and noise.Then after data aided signals filtered out, the pure interference signals and noise
are obtained. Define the pure interference signals and noise signals received by the mth antenna is
x� m (n), the new signals x� can be written as:
∑ x𝑚𝑚 (n)s𝑣𝑣𝐻𝐻 (n + τ�) (7)
x� 𝑚𝑚 (n) = x𝑚𝑚 (n) − s (n + τ�)
∑ s𝑣𝑣 (n + τ�)s𝑣𝑣𝐻𝐻 (n + τ�) 𝑣𝑣
The new antenna array siganls x� is:
x� 1 (8)
�=� ⋮ �
𝒙𝒙
x� 𝑀𝑀
By using the subspace spanned by the K + 1 largest principal eigenvectors of R = E{x� (t)x� H (t)},
a more accurate interference signals space estimator V �s can be calculated. As the signals
transmission rate is far faster than indoor position terminals, the interference signals space
�s calculated in train data time period is fitted to position data time period.
estimator V
Train Data Train Data
Signal Subspace Signal Subspace
Update Update
Figure 1: signal subspace update.
3.1. IPNCM
The new signals subspace V�s belong to Vs and more accurate. So applying the new signal subspace
H
�s V
projection matrix V �s will achieve a better performance:
� 𝒔𝒔 𝑯𝑯 𝐚𝐚(𝜃𝜃) = 𝐚𝐚�(𝜃𝜃) + 𝑒𝑒̃∥ (𝜃𝜃)
� 𝒔𝒔 𝑽𝑽
𝑽𝑽 (9)
Written the signals subspace as column vector:
� 𝑠𝑠 = [𝒗𝒗𝟏𝟏 ; ⋯ ; 𝒗𝒗K ]
𝑽𝑽 (10)
H
�s V
In(10), vi is the ith eigenvector. V �s a(θ) = a� (θ) + e� ∥ (θ) all can be linear represented by
signals subspace V �s :
K+1 (11)
𝐚𝐚�(𝜃𝜃) + 𝑒𝑒̃∥ (𝜃𝜃) = � 𝒌𝒌𝒊𝒊 (𝜃𝜃)𝒗𝒗𝒊𝒊
𝑖𝑖=1
In (11), k i (θ)are combination coefficient correspond to a� (θ) + e� ∥ (θ), and which are K × 1
column vector:
𝒌𝒌𝟏𝟏 (𝜃𝜃) (12)
𝑲𝑲(𝜃𝜃) = � ⋮ �
𝒌𝒌𝑲𝑲 (𝜃𝜃)
Then, the ideal DOA vector after subspace projection can be written as:
𝑽𝑽 � 𝒔𝒔 𝑯𝑯 𝑨𝑨 = 𝑽𝑽
� 𝒔𝒔 𝑽𝑽 � 𝒔𝒔 [𝑲𝑲(𝜃𝜃1 ) ⋯ 𝑲𝑲(𝜃𝜃𝐾𝐾 )] = 𝑽𝑽 � 𝒔𝒔 𝑲𝑲𝒌𝒌𝒌𝒌 (13)
The K × K matrix K kk are composed by combination coefficient column vectors. By using the he
ideal DOA vector after subspace projection V �s H A to calculate the interference-noise subspace,
�s V
the IPNCM algorithm can form nulls in the direction of interference signals more accurately:
𝐾𝐾
� 𝒔𝒔 𝑯𝑯 𝐚𝐚(𝜃𝜃𝑖𝑖 )(𝑽𝑽
� 𝒔𝒔 𝑽𝑽
𝑽𝑽 � 𝒔𝒔 𝑯𝑯 𝐚𝐚(𝜃𝜃𝑖𝑖 ))𝑯𝑯
� 𝒔𝒔 𝑽𝑽 (14)
𝑹𝑹𝑖𝑖+𝑛𝑛 = �
� � 𝑯𝑯 𝐚𝐚(𝜃𝜃𝑖𝑖 ))𝑯𝑯 𝑹𝑹
𝑖𝑖=1 (𝑽𝑽𝒔𝒔 𝑽𝑽𝒔𝒔
� −𝟏𝟏 𝑽𝑽 � 𝒔𝒔 𝑯𝑯 𝐚𝐚(𝜃𝜃𝑖𝑖 )
� 𝒔𝒔 𝑽𝑽
3.2. SOI DOA estimation
As the new interference signals subspace do not contain SOI, so the SOI DOA estimation using the
orthogonal subspace projection to calculate:
� 𝑠𝑠 𝑽𝑽
�(𝜃𝜃𝑠𝑠 ) = (𝑰𝑰 − 𝑽𝑽
𝒂𝒂 � 𝐻𝐻
𝑠𝑠 )𝒂𝒂(𝜃𝜃𝑠𝑠 ) (15)
4. Numerical simulations
Chose a linear antenna array with 10 sensors. Distance between adjacent sensors is λ/2, where λ
denotes the wavelength corresponding to the frequency of SOI. In far field exist 1 signal and 2
interferences, defined as s, j1 and j2 with azimuth and directions 71°, 112°and 151° respectively.
The signal verification data use simple {1, -1} sequence. The interference and noise all meet
Gaussian distribution. The JNR is 20dB. The number of verification data snapshots is 200 and the
totally number of snapshots is 600. The compare algorithm includes LCMV algorithm [7], Worst-
Case algorithm [12] with error range parameter 3, IPNCM algorithm [20], Huang’s algorithm [29]
with integrate number parameter 1024 and estimation error parameter 0.1, Yuan’s algorithm [23]
with subspace size parameter 0.9, Liu’s algorithm [27] with subspace size parameter 0.8, Sun’s
algorithm [32] with parameter η 0.1, Yang’s algorithm [35] with parameter ε 0.3 and ρ 0.9. All the
IPNCM-like algorithm with phase sector parameter 8°.
4.1. Mismatch by gain and phase error
Gain error meet Gaussian distribution (0,0.052) and phase error meet Gaussian distribution (0,
(0.025 π)2). Change the SNR from -10dB to 30dB, the SINR of SOI by the proposed robust IPNCM
algorithm and the others algorithm were calculated respectively across 200 Monte Carlo
simulations and shown in Figure 2.
40
OPT
LCMV
Worst-Case
30 IPNCM
Huang
Yuan
Liu
20
Sun
Yang
Proposed
10
SINR/dB
0
-10
-20
-10 -5 0 5 10 15 20 25 30
SNR/dB
Figure 2: SOI SINR by different algorithms var SNR.
In Figure 2, the proposed algorithm achieves a much more robust adaptive beamforming
performance and almost achieve the global optimal performance. Next, we exam the snapshot
number effect to algorithm performance. Fix the SNR as 15dB, change the snapshots number from
150 to 900, and the verification data is changed from 50 to 300, the SINR of SOI by the proposed
robust IPNCM algorithm and the others algorithm were calculated respectively across 200 Monte
Carlo simulations and shown in Figure 3. In Figure 3, the proposed algorithm achieve robust
adaptive beamforming performance while the snapshots are few.
30
25
20
15
OPT
SINR/dB
LCMV
10 Worst-Case
IPNCM
Huang
5 Yuan
Liu
Sun
0 Yang
Proposed
-5
200 300 400 500 600 700 800 900
Snapshots Number
Figure 3: SOI SINR by different algorithms var different snapshots number.
4.2. Mismatch by signal direction bias
Direction biases of three signals are all meet uniform distribution in section of ±0.02 π. Change the
SNR from -10dB to 30dB, the SINR of SOI by the proposed robust IPNCM algorithm and the others
algorithm were calculated respectively across 200 Monte Carlo simulations and shown in Figure 4.
40
30
20
OPT
LCMV
10 Worst-Case
IPNCM
Huang
Yuan
0
Liu
Sun
Yang
-10 Proposed
-20
-10 -5 0 5 10 15 20 25 30
Figure 4: SOI SINR by different algorithms var SNR.
In Figure 4, the proposed algorithm almost achieves the global optimal performance. Next, we
exam the snapshot number effect to algorithm performance. Fix the SNR as 15dB, change the
snapshots number from 150 to 900, and the verification data is changed from 50 to 300, the SINR of
SOI by the proposed robust IPNCM algorithm and the others algorithm were calculated
respectively across 200 Monte Carlo simulations and shown in Figure 5. In Figure 5, the proposed
algorithm will achieve a robust adaptive beamforming performance while the snapshots are few.
30
25
20
15
OPT
10
LCMV
SINR/dB
Worst-Case
5 IPNCM
Huang
Yuan
0
Liu
Sun
-5
Yang
Proposed
-10
-15
200 300 400 500 600 700 800 900
Snapshots Number
Figure 5: SOI SINR by different algorithms var different snapshots number.
4.3. Mismatch by array geometry error
Array sensor’s location errors are all meet uniform distribution in section of ±0.01λ. Change the
SNR from -10dB to 30dB, the SINR of SOI by the proposed robust IPNCM algorithm and the others
algorithm were calculated respectively across 200 Monte Carlo simulations and shown in Figure 6.
40
30
20
OPT
LCMV
10
Worst-Case
INPCM
Huang
0
Yuan
Liu
-10 Sun
Yang
Proposed
-20
-10 -5 0 5 10 15 20 25 30
Figure 6: SOI SINR by different algorithms var SNR.
In Figure 6, the proposed algorithm almost achieves the global optimal performance. Next, we
exam the snapshot number effect to algorithm performance. Fix the SNR as 15dB, change the
snapshots number from 150 to 900, and the verification data is changed from 50 to 300, the SINR of
SOI by the proposed robust IPNCM algorithm and the others algorithm were calculated
respectively across 200 Monte Carlo simulations and shown in Figure 7. In Figure 7, the proposed
algorithm achieves robust adaptive beamforming performance while the snapshots are few.
26
24
OPT
22 LCMV
Worst-Case
IPNCM
20 Huang
Yuan
SINR/dB
Liu
18 Sun
Yang
Proposed
16
14
12
150 300 450 600 750 900
Snapshots Number
Figure 7: SOI SINR by different algorithms var different snapshots number.
5. Conclusions
In array signal process, IPNCM is very helpful to interference filtered out as the SNR of SOI is often
greater than 0. In practical engineering applications, estimation error caused by gain error and
phase error are always exist and will cause SOI loss in IPNCM. The estimation error perpendicular
to signal subspace part can be filter out by using subspace algorithm. But the estimation error
parallel to signal subspace part still remains and effect the IPNCM performance.Aim at this special
problem to filter out parallel part, we proposed a new robust IPNCM type algorithm. With the
aiding of the verification data, a more precise IPNCM can be estimated. The new IPNCM-type
algorithm will achieve a much more robust performance to the others which is already proved by
simulation results.
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